As \(s\rightarrow 0\) , the following asymptotic expansion of terms in (REF )–(REF )
holds (see e.g. [1]}):
\(z\circ \phi _s =z +s \nabla z \Lambda +{\rm o}(s),\quad g\circ \phi _s =g +s \nabla g \Lambda +{\rm o}(s),\quad C\circ \phi _s =g +s\nabla C \Lambda +{\rm o}(s),\\\nabla \phi _s^{-1}\circ \phi _s =I -s\nabla \Lambda +{\rm o}(s),\quad E(\nabla \phi _s^{-1}\circ \phi _s, \tilde{u}) =\epsilon (\tilde{u})-s E(\nabla \Lambda , \tilde{u}) +{\rm o}(s),\\\omega ^{\rm d}_s =1 +s\, {\rm div} \Lambda +{\rm o}(s),\quad \omega ^{\rm b}_s =1 +s\, {\rm div}_{\tau _t} \Lambda +{\rm o}(s)\\\nu _{t+s}\circ \phi _s =\nu _{t} +s \nabla \nu _t \Lambda +{\rm o}(s),\quad [\![\tilde{u}]\!]_{\tilde{\tau }_{t+s}} =[\![\tilde{u}]\!]_{\tau _t}+s [\![\tilde{u}]\!]_{\nabla \tau _t \Lambda } +{\rm o}(s)\) 
